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DEFINITION OF SEQUENCE SYNONYM MOD

The splitting lemma states that if the above short exact sequence admits a morphism t: B → A such that t o f is the identity on A or a morphism u: C → B such that g o u is the identity on C, then B is a twisted direct sum of A and C. It is helpful to think of A as a subobject of B with f being the embedding of A into B, and of C as the corresponding factor object B/ A, with the map g being the natural projection from B to B/ A (whose kernel is exactly A). Furthermore, the image of f is equal to the kernel of g. Important are short exact sequences, which are exact sequences of the formīy the above, we know that for any such short exact sequence, f is a monomorphism and g is an epimorphism.

A consequence of these last two facts is that the sequence 0 → X → Y → 0 is exact if and only if the map from X to Y is an isomorphism.

if and only if that map is an epimorphism (onto).

Dually, the sequence B → C → 0 is exact at C if and only if the image of the map from B to C is all of C, i.e.

if and only if that map is a monomorphism (one-to-one). Here 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z/2 Z is given by reducing integers modulo 2. This sequence may also be written without using special symbols for monomorphism and epimorphism: This is an exact sequence because the image 2 Z of the monomorphism is the kernel of the epimorphism.

DEFINITION OF SEQUENCE SYNONYM MOD

Here the hook arrow indicates that the map 2⋅ from Z to Z is a monomorphism, and the two-headed arrow indicates an epimorphism (the map mod 2). The second operation forms an element in the quotient space, j = i mod 2. The first operation forms an element in the set of integers, Z, using multiplication by 2 on an element from Z j = 2 i. So in that case a short exact sequence would be written as follows: ExampleĬonsider the following sequence of abelian groups: If instead the objects are groups not known to be abelian, then multiplicative rather than additive notation is traditional, and the identity element - as well as the trivial group - is often written as "1" instead of "0". The placement of the 0's forces ƒ to be a monomorphism and g to be an epimorphism (see below). Where 0 represents the zero object, such as the trivial group or a zero-dimensional vector space. In this case, A is a subobject of B, and the corresponding quotient is isomorphic to C:Ī short exact sequence of abelian groups may also be written as an exact sequence with five terms: Where ƒ is a monomorphism and g is an epimorphism. The most common type of exact sequence is the short exact sequence. More generally, the notion of an exact sequence makes sense in any category with kernels and cokernels. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms. Note that the sequence of groups and homomorphisms may be either finite or infinite.Ī similar definition can be made for certain other algebraic structures. Of groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to the kernel of the next: meanings of sequence will be translated.In the context of group theory, a sequence Input a term sequence by either copy & post, drag & drop, or simply by typing in the search box. This page is an online lexical resource, contains a list of the sequence like words in a Hindi language in the order of the alphabets, and that tells you what they mean, in the same or other languages including English. Indian Official Languages Dictionary is significantly better than Google translation offers multiple meanings, alternate words list of sequence sequence phrases with similar meanings in Hindi हिन्दी, Hindi हिन्दी dictionary Hindi हिन्दी sequence translation sequence meaning sequence definition sequence antonym sequence synonym Hindi language reference work for finding synonyms, antonyms of sequence. Sequence in Hindi Hindi of translation of sequence Hindi meaning of sequence what is sequence in Hindi dictionary definition, antonym, and synonym of sequence

Sequence | Hindi dictionary translates English to Hindi and Hindi to English sequence words sequence phrases with sequence synonyms sequence antonyms sequence pronunciations.